1. WK15. Vertex Cover and Approximation Algorithm By Lin, Jr-Shiun Choi, Jae Sung

2. Vertex Cover Definition A set of vertices in an undirected graph where every edge connects at least one vertex. The vertex cover problem is to find a minimum size set and is NP-complete. (NIST)verticesundirected graphedgeNP-completeNIST

3. Vertex Cover Example Determine the smallest subset of vertex that “Cover” the graph on the right. 1 2 3 4 5

4. Vertex Cover Example Determine the smallest subset of vertex that “Cover” the graph on the right. ANS: { 1, 3, 4 } 1 2 3 4 5

5. Approximation Algorithm Minimization problem if F, C > 1 such that you can find a solution which is <= CA in polynomial time. then this is a approximation solution Maximization problem if F, C > = CA in polynomial time. then this is a approximation solution

6. Approximation Algorithm for Vertex Cover Algorithm 1. choose a edge A, kick out all edges which connect to 2 ends of A ( include A). 2. choose other edges and repeat step1 until all edges are kick out. vertex-cover that is atmost twice the size of an optimal cover (rmuhamma)rmuhamma

7. Euclidean TSP Example TempP Approx T MST

8. Euclidean TSP Let Optimal TSP = T Let MST = M (TSP=spanning tree that visits all vertex) M should be smaller than T T >=M TempP=2m Approx T <=2M<=2T Approx T <= (1+ X )T X is any small number, e.g. 0.000001

9. Approximation Algorithm for TSP TSP is general graph that can not be approximated. Claim: an algorithm can solve it. we will show that approximation algorithm can be used to solve the Hamiltonian cycle problem GT’<= CT

10. Approximation Algorithm for TSP G Add weight Approximation Algorithm CN+1 CN+1 <= CT 5N+1<=5T (N= # of Vertex) 26<=5T